(\partial)/(\partial x)(x/(y^2)sin(x/y))

解

\frac{\partial}{\partial x} (\frac{x}{y ^{2}} sin (\frac{x}{y}))

\frac{y sin ( \frac{x}{y} ) + x cos ( \frac{x}{y} )}{y ^{3}}

解答ステップ

\frac{\partial}{\partial x} (\frac{x}{y ^{2}} sin (\frac{x}{y}))

y

を定数として扱う

定数を除去:

(a \cdot f)^{'} = a \cdot f^{'}

= \frac{1}{y ^{2}} \frac{\partial}{\partial x} (x sin (\frac{x}{y}))

積の法則を適用:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = x, g = sin (\frac{x}{y})

= \frac{1}{y ^{2}} (\frac{\partial}{\partial x} (x) sin (\frac{x}{y}) + \frac{\partial}{\partial x} (sin (\frac{x}{y})) x)

\frac{\partial}{\partial x} (x) = 1

\frac{\partial}{\partial x} (sin (\frac{x}{y})) = \frac{cos ( \frac{x}{y} )}{y}

= \frac{1}{y ^{2}} 1 \cdot sin (\frac{x}{y}) + \frac{cos ( \frac{x}{y} )}{y} x

簡素化

\frac{1}{y ^{2}} 1 \cdot sin (\frac{x}{y}) + \frac{cos ( \frac{x}{y} )}{y} x : \frac{y sin ( \frac{x}{y} ) + x cos ( \frac{x}{y} )}{y ^{3}}

= \frac{y sin ( \frac{x}{y} ) + x cos ( \frac{x}{y} )}{y ^{3}}